Weyl chamber
WeylChamber
2002-12-05 13:00:47
2004-04-08 09:36:10
Definition
positive Weyl chamber
dominant weight
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Let $E$ be a Euclidean vector space, $R\subset E$ a root system, and
$R^+\subset R$ a choice of positive roots. We define the {\em
positive Weyl chamber} (relative to $R^+$) to be the closed set
$$\mathcal{C}=\{u\in E\mid (u,\alpha)\geq 0\text{ for all }\alpha\in
R^+\}.$$
A weight which lies inside the positive Weyl chamber is
called {\em dominant}.
The interior of $\mathcal{C}$ is a fundamental domain for the action
of the Weyl group on $E$. The image $w(\mathcal{C})$ of $\mathcal{C}$
under the any element $w$ of the Weyl group is called a {\em Weyl
chamber}. The Weyl group $W$ acts simply transitively on the set of
Weyl chambers.